. If a, b, and c are nonzero, find the points where the line intersects each of the coordinate planes. If$a=0,$ explain why the line is parallel to the y z-plane.

A lineand$a=0$

$\text{Consider the line}\mathcal{L}\text{determined by the equations}r\left(t\right)=\left(x_0+at,y_0+bt,z_0+ct\right)\text{,}$

Then,$r\left(t\right)=\left(x_0+at,y_0+bt,z_0+ct\right)$

$\text{So,}x=x_0+at,y=y_0+bt,z=z_0+ct$

When the line intersect in yz-plane the value of $x=0$

$\text{Now substitute}x=0\text{in}x=x_0+at\text{.}$

$0=x_0+at$

$-x_0=at$

$\Rightarrow \frac{-x_0}{a}=t$

$\text{Substitute}t=\frac{-x_0}{a}\text{in}r\left(t\right)=\left(x_0+at,y_0+bt,z_0+ct\right)\text{.}$

$r\left(\frac{-x_0}{a}\right)=\left(x_0+a·\frac{-x_0}{a},y_0+b·\frac{-x_0}{a},z_0+c·\frac{-x_0}{a}\right)$

$r\left(\frac{-x_0}{a}\right)=\left(x_0-x_0,y_0+b·\frac{-x_0}{a},z_0+c·\frac{-x_0}{a}\right)$

$r\left(\frac{-x_0}{a}\right)=\left(0,y_0-\frac{b{x}_0}{a},z_0-\frac{c{x}_0}{a}\right)$

$\text{Thus the point in}yz\text{-plane is}\left(0,y_0-\frac{b{x}_0}{a},z_0-\frac{c{x}_0}{a}\right)\text{.}$

When the line intersect in zx -plane the value of $y=0$

$\text{Now substitute}y=0\text{in}y=y_0+bt\text{.}$

$0=y_0+bt$

$0-y_0=y_0+bt-y_0$

$-y_0=bt$

$\Rightarrow t=\frac{-y_0}{b}$

$\text{Substitute}t=\frac{-y_0}{b}\text{in}r\left(t\right)=\left(x_0+at,y_0+bt,z_0+ct\right)\text{.}$

$r\left(\frac{-y_0}{b}\right)=\left(x_0+a·\frac{-y_0}{b},y_0+b·\frac{-y_0}{b},z_0+c·\frac{-y_0}{b}\right)$

$r\left(\frac{-y_0}{b}\right)=\left(x_0-\frac{a{y}_0}{b},0,z_0-\frac{c{y}_0}{b}\right)$

$\text{Thus the point in}zx\text{-plane is}\left(x_0-\frac{a{y}_0}{b},0,z_0-\frac{c{y}_0}{b}\right)\text{.}$

When the line intersects in xy -plane the value of $z=0$

$\text{Now substitute}z=0\text{in}z=z_0+ct$

$0=z_0+ct$

$-z_0=ct$

$\Rightarrow t=\frac{-z_0}{c}$

$\text{Substitute}t=\frac{-z_0}{c}\text{in}r\left(t\right)=\left(x_0+at,y_0+bt,z_0+ct\right)\text{.}$

$r\left(\frac{-z_0}{c}\right)=\left(x_0+a·\frac{-z_0}{c},y_0+b·\frac{-z_0}{c},z_0+c·\frac{-z_0}{c}\right)$

$r\left(\frac{-z_0}{c}\right)=\left(x_0+\frac{-a{z}_0}{c},y_0+\frac{-b{z}_0}{c},z_0-z_0\right)$

$r\left(\frac{-z_0}{c}\right)=\left(x_0-\frac{a{z}_0}{c},y_0-\frac{b{z}_0}{c},0\right)$

$\text{Thus the point in}xy\text{-plane is}\left(x_0-\frac{a{z}_0}{c},y_0-\frac{b{z}_0}{c},0\right)\text{.}$

Thus the points where the line intersects the coordinate planes $yz,zx,xy$ are,

$\left(0,y_0-\frac{b{x}_0}{a},z_0-\frac{c{x}_0}{a}\right)\left(x_0-\frac{a{y}_0}{b},0,z_0-\frac{c{y}_0}{b}\right)\left(x_0-\frac{a{z}_0}{c},y_0-\frac{b{z}_0}{c},0\right)$

$\text{When}a=0\text{, the equation is}r\left(t\right)=\left(x_0+0·t,y_0+bt,z_0+ct\right)\text{.}$

$r\left(t\right)=\left(x_0,y_0+bt,z_0+ct\right)$

The x component becomes zero in yz-plane.

Thus, when $a=0$the line is parallel to yz-plane.

Therefore, the required answer is,